Multilevel Monte Carlo for multi-dimensional SDEs

Transcription

1 Mutieve Monte Caro for muti-dimensiona SDEs Mike Gies Oxford University Mathematica Institute Oxford-Man Institute of Quantitative Finance MCQMC, Warsaw, August 16-20, 2010 Mutieve Monte Caro p. 1/25

2 Mutieve approach Given a scaar SDE driven by a Brownian diffusion ds(t) = a(s,t) dt + b(s,t) dw(t), to estimate E[P] where the path-dependent payoff P can be approximated by P using 2 uniform timesteps, we use E[ P L ] = E[ P 0 ] + L =1 E[ P P 1 ]. E[ P P 1 ] is estimated using N simuations with same W(t) for both P and P 1, Ŷ = N 1 N i=1 ( (i) P ) (i) P 1 Mutieve Monte Caro p. 2/25

3 Mutieve approach Using independent sampes for each eve, the variance of the combined estimator is [ L ] { L V Ŷ = N 1 V[ V, V P P 1 ], > 0 V[ P 0 ], = 0 =0 =0 and the computationa cost is proportiona to L =0 N h 1 Hence, the variance is minimised for a fixed computationa cost by choosing N to be proportiona to V h. Mutieve Monte Caro p. 3/25

4 MLMC Theorem Theorem: Let P be a functiona of the soution of a stochastic o.d.e., and P the discrete approximation using a timestep h = 2 T. If there exist independent estimators Ŷ based on N Monte Caro sampes, with computationa compexity (cost) C, and positive constants α 1 2,β,c 1,c 2,c 3 such that i) E[ P P] c 1 h α E[ P 0 ], = 0 ii) E[Ŷ] = E[ P P 1 ], > 0 iii) V[Ŷ] c 2 N 1 h β iv) C c 3 N h 1 Mutieve Monte Caro p. 4/25

5 MLMC Theorem then there exists a positive constant c 4 such that for any ε<e 1 there are vaues L and N for which the mutieve estimator L Ŷ = Ŷ, =0 [ (Ŷ ) ] 2 has Mean Square Error MSE E E[P] < ε 2 with a computationa compexity C with bound c 4 ε 2, β > 1, C c 4 ε 2 (og ε) 2, β = 1, c 4 ε 2 (1 β)/α, 0 < β < 1. Mutieve Monte Caro p. 5/25

6 Previous Work First paper (Operations Research, ) appied idea to SDE path simuation using Euer-Maruyama discretisation Second paper (MCQMC ) used Mistein discretisation for scaar SDEs improved strong convergence gives improved mutieve variance convergence Mutieve method is a generaisation of two-eve contro variate method of Kebaier (2005), and simiar to ideas of Speight (2009) and aso reated to mutieve parametric integration by Heinrich (2001) Mutieve Monte Caro p. 6/25

7 Numerica Anaysis If P is a Lipschitz function of S(T), the vaue of the underying at maturity, the strong convergence property ( [ E (ŜN S(T)) 2]) 1/2 = O(h γ ) impies that V[ P P] = O(h 2γ ) and hence V V[ P P 1 ] = O(h 2γ ). Therefore β = 1 for Euer-Maruyama discretisation, and β =2 for the Mistein discretisation. However, in genera, good strong convergence is neither necessary nor sufficient for good convergence for V. Mutieve Monte Caro p. 7/25

8 Numerics and Anaysis Euer Mistein option numerics anaysis numerics anaysis Lipschitz O(h) O(h) O(h 2 ) O(h 2 ) Asian O(h) O(h) O(h 2 ) O(h 2 ) ookback O(h) O(h) O(h 2 ) o(h 2 δ ) barrier O(h 1/2 ) o(h 1/2 δ ) O(h 3/2 ) o(h 3/2 δ ) digita O(h 1/2 ) O(h 1/2 og h) O(h 3/2 ) o(h 3/2 δ ) Tabe: V convergence observed numericay (for GBM) and proved anayticay (for more genera SDEs) Euer anaysis due to G, Higham & Mao (Finance & Stochastics, 2009) and Avikainen (Finance & Stochastics, 2009). Mistein anaysis due to G, Debrabant & Rößer Mutieve Monte Caro p. 8/25

9 Other work Yuan Xia, G jump-diffusion modes Syvestre Burgos, G Greeks Hoe, von Schwerin, Szepessy, Tempone adaptive discretisations Dereich, Heidenreich Lévy processes Hickerne, Müer-Gronbach, Niu, Ritter compexity anaysis Müer-Gronbach, Ritter paraboic SPDEs G, Reisinger paraboic SPDEs Teckentrup, Scheich, Ciffe, G eiptic SPDEs Barth, Schwab, Zoinger eiptic SPDEs Mutieve Monte Caro p. 9/25

10 Muti-dimensiona SDEs The Mistein scheme for muti-dimensiona SDEs is Ŝ i,n+1 = Ŝi,n + a i h + j b ij W j,n j,k, b ij S b k ( W j,n W k,n Ω jk h A jk,n ) where Lévy areas are defined as A jk,n = tn+1 t n (W j (t) W j (t n )) dw k (W k (t) W k (t n )) dw j O(h) strong convergence, but hard to simuate A jk O(h 1/2 ) strong convergence in genera if A jk omitted Mutieve Monte Caro p. 10/25

11 Discretisation error anaysis Suppose we ignore the Lévy area terms what is the resuting difference between coarse and fine path approximations? Let the coarse path approximation be Ŝ c n+1 = R(Ŝc n) and the fine path approximation be Ŝ f n+1 = R(Ŝf n) + g n so to eading order the difference D n Ŝf n Ŝc n satisfies D n+1 = R S D n + g n Mutieve Monte Caro p. 11/25

12 Discretisation error anaysis Using a Brownian Bridge construction in which W n+1/2 = 1 2 (W n + W n+1 + Z) where Z N(0,h c ), find that, to eading order, g i,n = 1 2 j,k, b ij S b k ( W j,n Z k,n W k,n Z j,n ) Note: g 0 for scaar appications, and for vector appications satisfying the commutativity conditions b ij S b k = b ik S b j, i,j,k Mutieve Monte Caro p. 12/25

13 Discretisation error anaysis W and Z are O( h) and independent = g n = O(h) but E[g n ] = 0 (to eading order) = D n = O( h) but E[ D n ] = 0 (to eading order) Haven t achieved anything yet reay just shown O( h) strong convergence when Lévy area is negected. (Best that can be achieved knowing just the discrete W Cark & Cameron, 1980) Now comes the new idea use antithetic variates in Brownian Bridge construction. i.e. construct a second fine path using Z n instead of Z n Mutieve Monte Caro p. 13/25

14 Antithetic treatment Since g n is inear in Z n, this impies that, to eading order, D (2) n = D (1) n Higher order terms in asymptotic error anaysis give D (1) n + D (2) n = O(h) If the payoff function f(s T ) is twice differentiabe then ( ) ( ) 1 2 f(ŝf(1) )+f(ŝf(2) ) f(ŝc ) 1 D(1) (2) 2 n + D n f (Ŝc ) ( (1) ( D n ) 2 (2) + ( D n ) 2) f (Ŝc ) = O(h) Mutieve Monte Caro p. 14/25

15 Antithetic treatment Hence, for the mutieve estimator on eve we use Ŷ = N 1 N n=1 ( 1 (n1) 2 P + ) (n2) P P (n) 1 and with V[Ŷ] = N 1 V V = O(h 2 ). This assumed the payoff function was twice differentiabe. For a put or ca option, more carefu anaysis near the strike gives V = O(h 3/2 ) sti enough to ensure the overa cost is O(ε 2 ). Mutieve Monte Caro p. 15/25

16 Numerica test Heston stochastic voatiity mode: ds = r S dt + v S dw 1, 0 < t < T, dv = κ(θ v) + ξ v dw 2, 0 < t < T, with T =1, S(0)=100, r=0.05, κ=1, θ=0.04, ξ=0.25 and correation ρ = 0.5. Integrating factor used for voatiity discretisation to improve accuracy with arge timesteps Mark Broadie European ca option with discounted payoff exp( rt) max(s(t) K, 0) with strike K =100. Mutieve Monte Caro p. 16/25

17 Numerica test og 2 variance P P P eve og 2 mean P 10 P P eve Mutieve Monte Caro p. 17/25

18 Numerica test 10 8 ε=0.005 ε=0.01 ε=0.02 ε=0.05 ε= Std MC MLMC N 10 6 ε 2 Cost eve accuracy ε Mutieve Monte Caro p. 18/25

19 Discontinuous payoffs Antithetic treatment doesn t hep with discontinuous payoffs: O( h) paths near enough to strike for fine and coarse paths to be on opposite sides these have O(1) difference in payoffs, so V[ P P 1 ] E[( P P 1 ) 2 ] = O( h) For scaar SDEs, use conditiona expectation one timestep before maturity: effectivey smooths payoff over O( h) very hepfu when Ŝf Ŝc = O(h) minima benefit when Ŝf Ŝc = O( h) Mutieve Monte Caro p. 19/25

20 Discontinuous payoffs 1 Payoff 0.5 O(h) S T O(h 1/2 ) For paths in smoothed region, if Ŝf Ŝc = O(h) then f (S) = O(h 1/2 ) = P P 1 = O(h 1/2 ) and hence V[ P P 1 ] = O(h 3/2 ) Mutieve Monte Caro p. 20/25

21 Discontinuous payoffs For muti-dimensiona SDEs, approximate the Lévy areas by sub-samping W(t) within each timestep Question: how many sub-sampes to use? too few and there s no significant benefit too many and the computationa cost is excessive what is optima? If each timestep is divided into M sub-intervas, error in each Lévy area approximation is O(hM 1/2 ) Hence, strong convergence error and Ŝf Ŝc are both O(h 1/2 M 1/2 ), assuming M h 1 Mutieve Monte Caro p. 21/25

22 Discontinuous payoffs Using antithetic treatment, for paths in smoothed region 1 (f(ŝf(1) ) ( ) 2 )+f(ŝf(2) ) f(ŝc ) 1 D(1) (2) 2 n + D n f (Ŝc ) ( (1) ( D n ) 2 (2) + ( D n ) 2) f (Ŝc ) = O(h 1/2 + M 1 ) If M 1 h 1/2, then doubing M doubes the cost per path, but reduces the variance by factor 4 good! Optimum is when M = O(h 1/2 ) Mutieve variance is O(h 3/2 ) and cost is O(h 1/2 ) per path; compexity anaysis shows overa cost is O(ε 2 (og ε) 2 ). Mutieve Monte Caro p. 22/25

23 Numerica test Heston mode for digita ca P = exp( rt)k 1 S(T)>K og 2 variance P P P eve og 2 mean P 10 P P eve Mutieve Monte Caro p. 23/25

24 Numerica test 10 8 ε=0.01 ε=0.02 ε=0.05 ε=0.1 ε= Std MC MLMC N 10 6 ε 2 Cost eve accuracy ε Mutieve Monte Caro p. 24/25

25 Concusions mutieve method being adapted to increasingy more chaenging appications for muti-dimensiona SDEs with Lipschitz payoffs, negecting the Lévy area terms in the Mistein scheme can sti give good decay of the mutieve variance if antithetic variates are used for discontinuous payoffs, the Lévy areas need to be approximated but sti get good decay of the variance Papers are avaiabe from: giesm/finance.htm Mutieve Monte Caro p. 25/25